THE ZERO-ELECTRON-MASS LIMIT IN THE HYDRODYNAMIC MODEL FOR PLASMAS

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1 THE ZERO-ELECTRON-MASS LIMIT IN THE HYDRODYNAMIC MODEL FOR PLASMAS GIUSEPPE ALÌ, LI CHEN, ANSGAR JÜNGEL, AND YUE-JUN PENG Abstract. The limit of vanishing ratio of the electron mass to the ion mass in the isentropic transient Euler-Poisson equations with perioic bounary conitions is prove. The equations consist of the balance laws for the electron ensity an current ensity for given ion ensity, couple to the Poisson equation for the electrostatic potential. The limit is relate to the low-mach-number limit of Klainerman an Maja. In particular, the limit velocity satisfies the incompressible Euler equations with amping. The ifference to the zero-mach-number limit comes from the electrostatic potential which nees to be controlle. This is one by a reformulation of the equations in terms of the enthalpy, higher-orer energy estimates an a careful use of the Poisson equation. AMS Classification. 35B25, 35L60, 35L65, 35Q35, 82D10. Keywors. Euler-Poisson system, limit of vanishing electron mass, incompressible Euler equations, energy estimates. 1. Introuction A couple of years ago, the last two authors starte a program to erive rigorously some asymptotic limits, namely the relaxation limit, the quasineutral limit an the zero-electronmass limit, in quasi-hyroynamic moels for plasmas. Whereas most of these limits coul be rigorously prove, the zero-electron-mass limit in the hyroynamic equations remaine unsolve. In this paper we fill this gap for the hyroynamic moel with given ion ensity. First, consier the (scale hyroynamic equations for the electron ensity n e with charge q e = 1, the ensity n i of the positively charge ions with charge q i = +1, the respective velocities v e, v i an the electrostatic potential φ, t n α + (n α v α = 0, α = e,i, m α t (n α v α + m α (n α v α v α + p α (n α = q α n α φ m α n α v α τ α, λ 2 φ = n i n e C(x for x, t > 0, where 1 an enotes the -imensional torus. The initial conitions are given by n α (, 0 = n I,α, v α (, 0 = v I,α in, α = e,i. Li Chen is partially supporte by National Natural Science Founation of China (NSFC, grant number Ansgar Jüngel acknowleges partial support from the Austrian Science Fun (FWF, grant P20214 an WK Differential Equations, the German Science Founation (DFG, grant JU 359/7, an the Austrian-Croatian Project of the Austrian Exchange Service (ÖAD. This research is part of the ESF program Global an geometrical aspects of nonlinear partial ifferential equations (GLOBAL. 1

2 2 GIUSEPPE ALÌ, LI CHEN, ANSGAR JÜNGEL, AND YUE-JUN PENG In the above equations, p α are the pressure functions, usually given by p α (n = c α n γα, n 0, where c α > 0 an γ α 1 are constants. In this work, we only assume that p α is smooth an strictly increasing. The function C(x moels fixe charge backgroun ions (oping profile. The (scale physical parameters are the particle mass m α, the relaxation time τ α an the Debye length λ. We assume that the value of the integral φx is fixe; for instance, φx = 0. In this paper we restrict ourselves to a situation in which the ion ensity is given, i.e., we wish to perform rigorously the limit m e 0 in the system (1 (2 (3 t n e + (n e v e = 0, m e t (n e v e + m e (n e v e v e + p e (n e = n e φ m e n e v e τ e, λ 2 φ = n e N for x, t > 0, where N = n i C is given. (In fact, we nee that N is a constant. The parameter m e is essentially the ratio of the electron mass to the ion mass (see, e.g., [14] for etails on the scaling. We assume that the ion is much heavier than the electron such that the limit m e 0 makes sense. The limit has the goal to achieve simpler moels containing the essential physical phenomena. We notice that in plasma physics, zero-electron-mass assumptions are wiely use (see, e.g., [9, 17]. Concerning the existence of solutions to the hyroynamic moel in the one-imensional or multi-imensional situation, either in the torus or in the whole space, we refer to [2] for precise references. Whereas in the one-imensional case, the existence of general globalin-time weak entropy solutions has been shown, there are only results for smooth solutions for small times or initial ata close to an equilibrium state in the multi-imensional case, which we consier here. The relaxation limit in the hyroynamic (Euler-Poisson equations to the rift-iffusion moel, which has been first stuie by Marcati an Natalini [21], has been solve in [10, 12, 13] for weak entropy solutions (also see [1]. The quasineutral limit in the hyroynamic moel has been analyze for transient smooth solutions by Corier an Grenier [4] in the one-imensional case an inepenently in [25, 29] in the multi-imensional case. We refer also to [8] for an analysis of the limit in Vlasov-Poisson equations an [6] for a combine quasineutral-relaxation limit. The limit for steay states has been consiere in [23, 24, 27]. The zero-electron-mass limit in the transient equations has been achieve only uner restrictive assumptions; see [7]. For steay states, we refer to [23]. These asymptotic limits have been also stuie in the rift-iffusion equations which are obtaine in the relaxation limit. The quasineutral limit has been prove in [5, 15, 30]. In [14] the zero-electron-mass limit in these equations coul be shown (which is easier than in the hyroynamic moel. We mention that such limits have been recently analyze in quasi-hyroynamic quantum moels; see [11, 16]. Before we present the main ieas of this paper, it is convenient to write the main part of the system (1-(2 in symmetric hyperbolic form. Setting n = n e, v = v e, p(n = p e (n e, an ε 2 = m e an introucing the enthalpy h = h(n e, efine by h (n = p (n/n an

3 ZERO-ELECTRON-MASS LIMIT IN THE HYDRODYNAMIC MODEL 3 h(1 = 0, as a new variable, the system (1-(3 can be rewritten as (4 ( t + v h + p (n v = 0, ε 2 ( t + v v + h = φ ε 2 v, φ = n(h N, x, t > 0, where we assume that φx = 0, with initial conitions (5 h(, 0 = h ε I, v(, 0 = v ε I in. Here, we have set τ e = λ = 1 in orer to simplify the notation. Clearly, for smooth solutions, this system is equivalent to (1-(3. As we suppose that the pressure function is invertible, so oes h(n an we enote its inverse by n(h. The objective of this paper is to perform the limit ε 0 in ( Formal asymptotic analysis. Assume that N > 0 is a constant. In orer to erive the limiting system when ε 0, we substitute the formal expansions h = h 0 + εh 1 + ε 2 h 2 +, v = v 0 + εv 1 + ε 2 v 2 +, φ = φ 0 + εφ 1 + ε 2 φ 2 + in the system (4 an equate equal powers of ε. The lowest-orer terms satisfy the equations (6 ( t + v 0 h 0 + p (n(h 0 v 0 = 0, (h 0 φ 0 = 0, φ 0 = n(h 0 N. The secon equation implies that h 0 φ 0 is a function of time only. Combining this fact with the thir equation, we fin that h 0 solves h 0 = n(h 0 N. Employing the assumption φ 0 x = 0, it is not ifficult to see that φ 0 = 0 an h 0 = h(n R are the unique solutions of the corresponing equations, such that h 0 x = meas( h(n. In particular, the first equation in (6 becomes v 0 = 0. The first-orer terms satisfy (h 1 φ 1 = 0, φ 1 = n (h 0 h 1. The solutions h 1 = φ 1 = 0 are consistent with these equations. At secon orer, we fin (7 ( t + v 0 v 0 + v 0 = (φ 2 h 2, φ 2 = n (h 0 h 2. From v 0 = 0 an the first equation, v 0 an φ 2 h 2 can be foun. Then, h 2 is the solution of the thir equation, written in the form h 2 = n (h 0 h 2 (φ 2 h 2 an finally, φ 2 is given by φ 2 = h 2 +(φ 2 h 2. These consierations motivate to choose the initial ata as (8 h ε I = h 0 I + ε 2 h 2 I, v ε I = v 0 I + εv 1 I. The formal analysis shows that the zero-electron-mass limit has some similarities with the low-mach-number limit in the compressible Euler system [20]. It is possible to use ieas from Klainerman an Maja [18, 19] to eal with the term ε 1 h in (4 (after ivision by ε. However, we have another singularity from ε 1 φ which cannot be fixe by their metho. Our iea is to control this term by a careful use of the mass conservation an the Poisson equation. To escribe the iea more precisely, introuce the new variables h = h h0 φ φ, φ 0 = ε ε

4 4 GIUSEPPE ALÌ, LI CHEN, ANSGAR JÜNGEL, AND YUE-JUN PENG as in [20, Ch. 2.4], where φ 0 is any constant fixe by φx (if φx = 0 then φ 0 = 0. The system (4 can be written as (9 (10 (11 A(ε h( t + v h + 1 ε v = 0, ( t + v v + 1 ε h = 1 ε φ v, φ = 1 ε (n(ε h + h 0 n(h 0, x, t > 0, where A(ε h = 1/p (ε h + h Main ieas. For the proof of the limit ε 0 we nee to erive uniform estimates up to sth-orer erivatives with s > / Here, we will escribe only how to erive the lowest-orer estimates, which is sufficient to illustrate the iea. We assume that there are L (0,T;W 1, ( estimates for h an v. Frierich s energy estimate for symmetric hyperbolic systems an integration by parts yiel (A(ε h h 2 + v 2 x + v 2 x c ( h 2 + v 2 x 1 φ vx, t T ε where the constant c > 0 epens on the L (0,T;W 1, ( bouns for h an v. Replacing the term ε 1 v by the mass conservation equation, we are left to control the integrals φa(ε h ht x + φa(ε hv hx. The secon integral can be easily controlle (after integration by parts by the L (0,T; W 1, ( bouns for h an v. In orer to eal with the first integral we employ the Poisson equation, φ t = n (ε h + h 0 h t. Then we arrive at φa(ε h ht x = A(ε h n (ε h + h 0 φ t φx. Again after integration by parts, we obtain an integral with a goo sign, t φ 2 L 2 an other integrals which can be estimate by ε φ t L 2 an t (n (ε h + h 0 L 2. Using the Poisson equation to boun the first expression, it can be seen that both terms contain the erivative ε h t as above but now incluing the factor ε. Inee, by (9, we are now able to control this expression in some norm in terms of the L (0,T;W 1, ( estimates for h an v. For higher-orer erivatives, we nee to take care of the nonlinear terms arising from the partial erivatives, but finally, we en up with estimates for h an v which are appropriate to employ the stanar continuation argument (see below for etails. We remark that a more general strategy to perform the low-mach-number limit, an possibly also the zero-electron-mass limit, has been suggeste by Métivier an Schochet in [22]. However, the technique of Klainerman an Maja is quite funamental an sufficient

5 ZERO-ELECTRON-MASS LIMIT IN THE HYDRODYNAMIC MODEL 5 for our purpose. For other results on the incompressible limit of the compressible Euler equations, we refer to [3, 28] Main results. In orer to formulate our main theorems, we introuce as in [20] the following notations: s = H s (, s,t = sup s for s R, = L (T. 0<t<T Theorem 1. Let n be a smooth strictly increasing function an let N > 0. Furthermore, let s > /2 + 1 an let the initial ata (h ε I,vε I satisfy vε I = v0 I an h ε I h0 ε + vi ε s M 0, s where h 0 = h(n an M 0 > 0 is a constant inepenent of ε. Then there exist constants T 0 > 0 an M 0 > 0, inepenent of ε an ε 0 (M 0 > 0 such that for all 0 < ε < ε 0 (M 0, the problem (4-(5 has a classical solution (h ε,v ε,φ ε in [0,T 0 ] satisfying h ε h 0 ε + v ε s,t0 + φ ε s,t0 ε M 0. s,t0 Theorem 2. Let the assumptions of Theorem 1 hol with vi 0 = 0 an (12 h ε I h0 ε 2 M 1. s Let (h ε,v ε,φ ε be a classical solution to (4-(5 in [0,T 0 ] with T 0 > 0 inepenent of ε. Then, as ε 0, h ε h 0, φ ε 0 strongly in L (0,T 0 ;H α ( C 0,1 ([0,T 0 ];L 2 (, v ε v 0 strongly in C 0 ([0,T 0 ];H α ( for all α < s, where v 0 is the (unique classical solution of the following incompressible Euler equations with amping, (13 v 0 = 0, ( t + v 0 v 0 + v 0 = π, x, t > 0, v 0 (, 0 = v 0 I in, an π is the limit of ( φ ε h ε π weakly* in L (0,T 0 ;L 2 (. ε The paper is organize as follows. In section 2, uniform estimates are shown an Theorem 1 is prove by means of a continuation argument. Section 3 is evote to the proof of Theorem 2.

6 6 GIUSEPPE ALÌ, LI CHEN, ANSGAR JÜNGEL, AND YUE-JUN PENG 2. Uniform local existence First we recall for convenience some Moser-type inequalities which we will use in the subsequent analysis. Let α = (α 1,...,α be a multi-inex. The ifferential operator D α is efine by D α = α 1 x 1 α x ; D s for s N enotes the sth erivative. Let s 0, f, g H s ( L (, an α a multi-inex with α s. Then, for some constant c s > 0, (14 D α (fg 0 c s ( f D s g 0 + g D s f 0. Let s 1, f H s ( with Df L (, g H s 1 ( L ( an α s. Then, for some constant c s > 0, (15 D α (fg fd α g 0 c s ( Df D s 1 g 0 + g D s f 0. For the proof of Theorem 1 we employ the basic theory of smooth solutions to hyperbolic systems of Maja [20]. The key result is containe in the following lemma. Lemma 3. Suppose that it hols, for some T > 0 (maybe epening on ε an M > 0 (inepenent of ε, (16 h L (0,T ;W 1, ( + v L (0,T ;W 1, ( M. Then there exist ε 0 = ε 0 (M > 0 an c(m > 0 such that for all 0 < ε < ε 0, it hols (17 h s,t + v s,t + φ s,t e c(mt (M 0 + c(mt. We assume Lemma 3 for the moment an procee with the proof of Theorem 1. Proof of Theorem 1. In view of the existence results of [20], the proof follows from a continuation argument. We follow the lines of [20, p. 59]. First, let M 1 > M 0 an fix T 1 > 0 such that e c(m 1T 1 (M 0 + c(m 1 T 1 M 1. Writing φ = 1 (n(h N (with perioic bounary conitions an using the properties of the linear operator 1, the first two equations in (4 form a symmetric hyperbolic system with the unknowns (h, v. Therefore, the local existence results of [20] show that there exists a local smooth solution (h,v. Let T(ε be the maximal time of existence of such a smooth solution. If T(ε = + for all ε > 0, we are one. Otherwise, T(ε < + for some ε > 0 an then There exists t 1 (ε < T(ε such that lim sup( h s,t + v s,t + φ s,t = +. t T(ε (18 h s,t1 (ε + v s,t1 (ε + φ s,t1 (ε M 1 an (19 h s,t + v s,t + φ s,t > M 1 for all t > t 1 (ε. Lemma 3, applie with some T > 0 an M = M 1, provies the existence of ε 0 > 0 (inepenent of T such that (17 hols for all ε < ε 0. If t 1 (ε T 1 for all 0 < ε < ε 0, the

7 ZERO-ELECTRON-MASS LIMIT IN THE HYDRODYNAMIC MODEL 7 proof is finishe. Otherwise, we argue by contraiction; there exists 0 < ε 1 < ε 0 such that t 1 (ε 1 < T 1. In particular, h s,t1 (ε 1 + v s,t1 (ε 1 + φ s,t1 (ε 1 M 1. Now, we apply Lemma 3 with T = t 1 (ε 1 an M = M 1. This oes not change the value of ε 0 since it only epens on M 1. Then, by (17, h s,t1 (ε 1 + v s,t1 (ε 1 + φ s,t1 (ε 1 e c(m 1t 1 (ε 1 (M 0 + c(m 1 t 1 (ε 1 < e c(m 1T 1 (M 0 + c(m 1 T 1 M 1. Thanks to the strict inequality sign, we may exten, again by local existence results, the time interval [0,t 1 (ε 1 ] to [0,t 2 (ε 1 ] for some t 2 (ε 1 > t 1 (ε 1 such that h s,t2 (ε 1 + v s,t2 (ε 1 + φ s,t2 (ε 1 M 1. But this contraicts the efinition of t 1 (ε. Hence, t 1 (ε T 1 > 0 for all 0 < ε < ε 0, which shows that there exists a solution on [0,T 1 ] an T 1 oes not epen on ε. Proof of Lemma 3. Step 1: preparations. First we collect some useful inequalities which we will employ several times in this proof. Let f be a smooth function an let (16 hol. Then there exist constants c(m, c 0 (M, c 1 (M > 0 such that (20 0 < c 0 (M f (m (ε h c 1 (M for all m N an sufficiently small ε, sup f(ε h εc(m, (0,T sup t f(ε h c(m. (0,T In fact, the first estimates can be erive irectly by elementary computations. The last one is a consequence of the first, using (16 an (9, t f(ε h,t f (ε h,t ε t h,t c(m( εv h,t + A 1 (ε h v,t c(m. We will use the following notations. Let α with α s be a multi-inex. Then we efine D α u = sup α D α u. Furthermore, we abbreviate h α = D α h, vα = D α v, φ α = D α φ. Step 2: Frierich s energy estimates. We ivie (9 by A(ε h, apply the operator D α an multiply by A(ε h. Differentiating also (10 an (11, the resulting equations are (21 (22 (23 A(ε h t h α + A(ε hv h α + 1 ε v α = F α, t v α + v v α + 1 ε h α = 1 ε φ α v α + G α, φ α = n (ε h + h 0 h α + H α,

8 8 GIUSEPPE ALÌ, LI CHEN, ANSGAR JÜNGEL, AND YUE-JUN PENG where F α = A(ε h(v h α D α (v h + 1 ε( vα A(ε hd α (A 1 (ε h v, G α = v v α D α (v v, (24 H α = 1 ε Dα (n(ε h + h 0 n (ε h + h 0 h α. Frierich s energy estimate for (21-(22 (i.e., multiplying (21 by h α an (22 by v α, integrating over, taking the sum an integrating by parts give 1 ( A(ε h hα 2 + v α 2 x + v α 2 x 2 t T B ( h α 2 + v α 2 x + ( F α 2 + G α 2 x + 1 (25 φ α v α x T ε = I 1 + I 2 + I 3, where B = t (A(ε h,t + A(ε h,t v,t + A(ε h,t v,t + c v,t + 1. Step 3: control of I 1, I 2 an I 3. The inequalities (20 show that I 1 c(m ( h α 2 + v α 2 x. The integral I 2 can be estimate in a similar way as one by Klainerman an Maja [18, 19]. For convenience, we present the etails. We employ Moser-type estimates (14- (15 to obtain for α 1 (noting that F 0 = G 0 = 0 F α 0 A(ε h v h α D α (v h ε v α A(ε hd α (A 1 (ε h v 0 ( c A(ε h v D α 1 h 0 + D α v 0 h + c ( A(ε h D α 1 (A 1 (ε h v 0 + D α A(ε h 0 A 1 (ε h v ε ( c(m ( D α h 0 + D α v 0 + c(m A 1 (ε h D α v 0 + v D α 1 A 1 (ε h ε D α A(ε h 0 (26 an ( c(m D α h 0 + D α v G α 0 v v α D α (v v 0 c ( v D α 1 v 0 + D α v 0 v c(m ( D α v

9 ZERO-ELECTRON-MASS LIMIT IN THE HYDRODYNAMIC MODEL 9 This yiels ( I 2 c(m D α h D α v Now we turn to the elicate integral I 3. This term cannot be controlle irectly from the Riesz transformation (erive from the coupling with the Poisson equation, as one in [2], for instance. Our iea is to replace the singular term ε 1 φ α, after integration by parts in I 3, by equation (21, I 3 = 1 φ α v α x ε T = φ α F α x + φ α A(ε hv h α x + φ α A(ε h t h α x = K 1 + K 2 + K 3. Step 4: control of K 1, K 2 an K 3. With the help of the estimate (26, it is not ifficult to see that the first integral K 1 can be controlle by K 1 1 ( φ α 2 + F α 2 x c(m ( φ α 2 + D α h 2 + D α v 2 x + c(m. 2 We integrate by parts in the secon integral since h α cannot be estimate by D α h, K 2 = φ α va(ε hh α x φ α A(ε h vh α x c(m ( φ α 2 + φ α 2 + h α 2 x. φ α A(ε h vh α x The ifficult term is now K 3. In orer to control K 3, we cannot use (21 since this woul (again give a term containing ε 1 v α. Our strategy is to employ the Poisson equation in the following way. Taking the time erivative of (23, we have t φ α = n (ε h + h 0 t h α + t (n (ε h + h 0 h α + t H α, recalling the efinition (24 of H α. Then, multiplying this equation by φ α A(ε h/n (ε h+h 0 an substituting the expression into K 3, we arrive at K 3 = A(ε h n (ε h + h 0 φ α t φ α x A(ε h n (ε h + h 0 φ α t H α x = L 1 + L 2 + L 3. A(ε h n (ε h + h 0 φ α t (n (ε h + h 0 h α x

10 10 GIUSEPPE ALÌ, LI CHEN, ANSGAR JÜNGEL, AND YUE-JUN PENG Step 5: control of L 1 an L 2. After integration by parts we obtain L 1 = 1 ( A(ε h 2 n (ε h + h 0 t φ α 2 A(ε h x t φ α φ α x n (ε h + h 0 = 1 A(ε h 2 t n (ε h + h 0 φ α 2 x + 1 ( A(ε h t φ α 2 x 2 n (ε h + h 0 ( A(ε h t φ α φ α x. n (ε h + h 0 The boun (20 allows to estimate the secon an thir integral: L 1 1 A(ε h 2 t n (ε h + h 0 φ α 2 x+c(m φ α 2 x+c(m ( ε t φ α 2 + φ α 2 x. In orer to estimate the integral over ε t φ α 2, we employ again the Poisson equation (23, now written in the form For α = 0, we have, observing (20, t φ α = 1 ε Dα t (n(ε h + h 0. ε t φ 0 c t (n(ε h + h 0 0 c(m. For 1 α s, we procee by inuction. Elliptic estimates give ε t φ α 0 c ( D α t (n(ε h + h ε t φ α 0 c ( D α 1 (n (ε h + h 0 ε t h 0 + ε t φ α 0. The last term is boune by the inuction hypothesis. In orer to boun the first term, we employ (21. This gives controllable terms since the time erivative provies the factor ε in front of h t an ε h t can be estimate. Therefore, by Moser-type calculus, ε t φ α 0 c(m( D α h 0 + D α v Hence, the integral L 1 is boune by L 1 1 T A(ε h φ α 2 x + c(m ( D α h 2 + D α v 2 + φ α 2 + φ α 2 x + c(m. 2 t n (ε h + h 0 The estimate of L 2 uses again the fact that the term ε t h can be controlle, employing (21. More precisely, we have L 2 c(m t (n (ε h + h 0 ( h α T 2 + φ α 2 x c(m ( h α 2 + φ α 2 x.

11 ZERO-ELECTRON-MASS LIMIT IN THE HYDRODYNAMIC MODEL 11 Step 6: control of L 3. First we write, recalling the efinition (24 of H α, ( A(ε h 1 L 3 = n (ε h + h 0 φ α t ε Dα (n(ε h + h 0 n (ε h + h 0 h α x. Before we escribe our iea how to eal with this terms, we notice that H α = 0 for all α with α = 1 an thus L 3 = 0. Therefore, we may assume that 2 α s. The natural iea is to take the time erivative in the above integral an then to employ Moser-type inequalities. However, in this case we woul obtain terms like t h which cannot be controlle by M. Our iea is to reformulate the integral in such a way that only terms like ε h t an not h t appear. For convenience, we set g(ε h = A(ε h/n (ε h + h 0. Then, by a slight abuse of notation for D α, a reformulation gives ( 1 L 3 = g(ε hφ α t T ε Dα 1 (n (ε h + h 0 εd h D α (n (ε h + h 0 h x g(ε hφ α t (D α (n (ε h + h 0 h n (ε h + h 0 h α x T ( = g(ε hφ α t D α 1 (D(n (ε h + h 0 hx g(ε hφ α D α t (n (ε h + h 0 h n (ε h + h 0 D α t h t (n (ε h + h 0 h α x. We write these two integrals in the following way, using integration by parts: g(ε hφ α t D α 1 (D(n (ε h + h 0 hx ( = g(ε hφ α D α 1 t (Dn (ε h + h 0 h + Dn (ε h + h 0 t h x T = D(g(ε hφ α D α 2 ( t (Dn (ε h + h 0 hx T + g(ε hφ α D α 1( Dn (ε h + h 0 t h x = N1 + N 2 an g(ε hφ α (D α t (n (ε h + h 0 h n (ε h + h 0 D α t h t (n (ε h + h 0 h α x ( = D(g(ε hφ α D α 1 t (n (ε h + h 0 h n (ε h + h 0 D α 1 t h x g(ε hφ α Dn (ε h + h 0 D α 1 t hx + g(ε hφ α t (n (ε h + h 0 h α x T ( = D(g(ε hφ α D α 1 (n (ε h + h 0 t h n (ε h + h 0 D α 1 t h x + D(g(ε hφ α D α 1 ( t (n (ε h + h 0 hx g(ε hφ α Dn (ε h + h 0 D α 1 t hx

12 12 GIUSEPPE ALÌ, LI CHEN, ANSGAR JÜNGEL, AND YUE-JUN PENG + g(ε hφ α t (n (ε h + h 0 h α x = N 3 + N 4 + N 5 + N 6. Before we estimate N 1,...,N 6, we notice the following useful inequalities. Let f be a smooth function an 2 α s. Then, by Moser-type calculus (14 an employing (21, (27 D α 1 t f(ε h 0 = D α 1 (f (ε hε t h 0 c ( f (ε h ε t h α ε t h D α 1 f (ε h 0 c(m ( ε h α v α εf α h α 1 0 c(m( D α h 0 + D α v Furthermore, by applying Gagliaro-Nirenberg s inequality, it is not ifficult to verify that (28 D α f(ε h 0 εc( h D α h 0. For the term N 1 we use first integration by parts an then (14 an (20: ( N 1 = D(g(ε hφ α D α 1 ( t (n (ε h + h 0 h D α 2 ( t (n (ε h + h 0 D h x = N 4 + D(g(ε hφ α D α 2 ( t (n (ε h + h 0 D hx N 4 + (g(ε hφ α 0 D α 2 ( t (n (ε h + h 0 D h 0 N 4 + c(m( εφ α 0 + φ α 0 ( t n (ε h + h 0 h α h D α 2 t n (ε h + h 0 0 N 4 + c(m ( φ α 2 + φ α 2 + D α h 2 + D α v 2 x + c(m, where in the last inequality we have employe (27. In a similar way, using (28, we obtain N 2 c(m φ α 0 D α 1 (Dn (ε h + h 0 t h 0 ( c(m φ α 0 Dn (ε h + h 0 D α 1 t h 0 + t h D α n (ε h + h 0 0 ( c(m φ α 0 εd α 1 t h 0 + ε t h D α h 0 c(m ( φ α 2 + D α h 2 + D α v 2 x + c(m, since the integral over ε t h α 1 can be boune. The thir integral N 3 is estimate by means of (15: N 3 ( D(g(ε hφ α 0 Dn (ε h + h 0 D α 2 t h 0 + t h D α 1 n (ε h + h 0 0 c(m( εφ α 0 + φ α 0 ( εd α 2 t h 0 + t h εd α 1 h 0 c(m ( φ α 2 + φ α 2 + D α h 2 + D α v 2 x + c(m.

13 ZERO-ELECTRON-MASS LIMIT IN THE HYDRODYNAMIC MODEL 13 Here, the conition α 2 is essential to obtain, after ifferentiation, terms like ε t h instea of t h only. The remaining integrals N5 an N 6 are estimate in a similar way: N 5 c(m φ α 0 Dn (ε h + h 0 D α 1 t h 0 c(m ( φ α 2 + D α h 2 + D α v 2 x + c(m, T N 6 c(m ( φ α 2 + h α 2 x. Summarizing, we have foun that L 3 c(m ( φ α 2 + φ α 2 + D α h 2 + D α v 2 x + c(m. Step 7: en of the proof. The bouns for L 1, L 2 an L 3 in Steps 5 an 6 show that the integral K 3 from Step 4 can also be controlle. The control of K 1, K 2 an K 3 then controls I 3 from Step 3. Finally, the bouns for I 1, I 2 an I 3 give the inequality (see (25 ( 1 A(ε h h α 2 + v α 2 A(ε h + 2 t 2n (ε h + h 0 φ α 2 x + v α 2 x T c(m ( D α h 2 + D α v 2 + φ α 2 + φ α 2 x + c(m. Summing up over all multi-inices α with the same norm gives ( 1 A(ε h D α h 2 + D α v 2 A(ε h + 2 t 2n (ε h + h 0 D α φ 2 x + D α v 2 x T ( c(m ( D α h 2 + D α v 2 + D α φ 2 + φ 2 x + 1. By Gronwall s inequality, we obtain sup ( h,v, φ(,t s + v L 2 (0,T ;H s ( (M 0 + 1e c(mt 1 0<t<T e c(mt (M 0 + c(mt. This gives the assertion of the Lemma. We also nee estimates for the time erivative of h, v an φ. Lemma 4. Let the assumptions in lemma 3 an (12 hol, vi 0 ε 1 (0,ε 0 such that for all 0 < ε < ε 1, it hols = 0, then there exists h t 0,T + v t 0,T + φ t 0,T c(m,m 0,M 1,T.

14 14 GIUSEPPE ALÌ, LI CHEN, ANSGAR JÜNGEL, AND YUE-JUN PENG Proof. Taking the time erivative of (9-(11, we obtain (29 (30 (31 A(ε h t ht + A(ε hv h t + 1 ε v t = F t, t v t + v v t + 1 ε h t = 1 ε φ t v t + G t, φ t = n (ε h + h 0 h t, where F t = A(ε h(v h t t (v h + 1 ε( vt A(ε h t (A 1 (ε h v, G t = v v t t (v v = v t v. Frierich s estimates for (29-(30 give 1 (A(ε h h t 2 + v t 2 x + v t 2 x 2 t T ( F t 2 + G t 2 x + c(m ( h t 2 + v t 2 x + 1 φ t v t x. T ε The terms F t an G t can be easily controlle by F t 0 A(ε h ( h v t 0 + c(m v h t 0 c(m( h t 0 + v t 0, G t 0 v v t 0 c(m v t 0. The only elicate integral in the Frierich s estimate is the term involving 1/ε. In orer to control it we use (29: 1 φ t v t x = 1 φt v t x ε T ε T = φt F t x + φt A(ε hv h t x + φt A(ε h h tt x = P 1 + P 2 + P 3. The above estimate for F t gives P 1 c(m ( φ t 2 + h t 2 + v t 2 x. In orer to avoi the term h t in P 2, we integrate by parts: ( P 2 = φ t A(ε hv + φ t A(ε hv + φ t A(ε h v ht x c(m ( φ t 2 + φ t 2 + h t 2 x.

15 ZERO-ELECTRON-MASS LIMIT IN THE HYDRODYNAMIC MODEL 15 For the estimate of the remaining integral P 3, we take the time erivative of (31 an multiply the resulting equation by φ t A(ε h/n (ε h + h 0. This yiels A(ε h φ t htt = A(ε h φ t φ tt A(ε h n (ε h + h 0 n (ε h + h 0 t(n (ε h + h 0 φ t ht. Substituting the prouct A(ε h φ t htt in P 3 by the above expression, we obtain, setting g(ε h = A(ε h/n (ε h + h 0, P 3 = g(ε h φ t φ tt x g(ε h t (n (ε h + h 0 φ t ht x 1 g(ε h t φ t 2 x g(ε h φ tt φt x 2 + c(m t n (ε h + h 0 φ t ht x 1 g(ε h φ t 2 x + 1 t g(ε h φ t 2 x 2 t T 2 T + c(m ( φ t 2 + h t 2 + ε φ tt 2 x 1 g(ε h φ t 2 x + c(m ( φ t 2 + φ t 2 + h t 2 + ε φ tt 2 x. 2 t We employ again the Poisson equation to eal with the term ε φ tt. From (29 we fin that φ tt = t (n (ε h + h 0 h t + n (ε h + h 0 h tt = t (n (ε h + h 0 h t n (ε h + h 0 v h t 1 n (ε h + h 0 v t + n (ε h + h 0 F t. ε A(ε h A(ε h Multiplying this equation by ε 2 φtt, integrating over an integrating by parts yiels ε φ tt 2 x = ε 2 t (n (ε h + h 0 h t φtt x + ε 2 n (ε h + h 0 v h t φtt x T n (ε h + h 0 + ε v t φtt x ε 2 n (ε h + h 0 F t φtt x A(ε h A(ε h = ε 2 t (n (ε h + h 0 h t φtt x ε 2 (n (ε h + h 0 v h t φtt x T ( ε T 2 n (ε h + h 0 n (ε h + h 0 h t v φ tt x + ε v t φtt x n (ε h + h 0 + ε A(ε h v t φ tt x ε 2 A(ε h n (ε h + h 0 F t φtt x A(ε h

16 16 GIUSEPPE ALÌ, LI CHEN, ANSGAR JÜNGEL, AND YUE-JUN PENG 1 ε φ tt 2 x + c(m ( h t 2 + v t 2 x, 2 where we have use Poincaré s inequality φ tt 2 x c φ tt 2 x, which is allowe since the integral over φ an hence also over φ tt vanishes. Thus, the estimate of P 3 becomes P 3 1 g(ε h φ t 2 x + c(m ( φ t 2 + h t 2 + v t 2 x. 2 t We en up with an estimate of ε 1 φ t v t x, an thus, we obtain ( 1 A(ε h h t 2 + v t 2 A(ε h + 2 t 2n (ε h + h 0 φ t 2 x + v t 2 x T c(m ( φ t 2 + h t 2 + v t 2 x. The proof of the lemma is complete by application of Gronwall s lemma if a boun for the initial ata is available, i.e. ( h t,v t, φ t (, 0 0 c(m 0. In fact, from (9 an using assumption vi ε = v0 I with v0 I = 0, we have h t (, 0 0 c(m 0 ( (h ε I h v 1 I 0 c(m 0, an similarly for v t (, 0 with the help of assumption (12 an φ t (, 0 0 C h t (, Proof of Theorem 2 Let (h ε,v ε,φ ε be a (classical solution to (4-(5 in the interval [0,T 0 ] with T 0 inepenent of ε. The estimates of Lemmas 3 an 4 show that the following uniform bouns hol: ε 1 (h ε h 0 s,t0 + v ε s,t0 + ε 1 φ ε s,t0 M, The inequalities imply that, as ε 0, ε 1 h ε t 0,T0 + v ε t 0,T0 + ε 1 φ ε t 0,T0 c(m. h ε h 0, φ ε 0 strongly in L (0,T 0 ;H s ( C 0,1 ([0,T 0 ];L 2 (. Furthermore, by Aubin s lemma (see Theorem 5 in [26], there exists a subsequence of v ε, which is not relabele, such that v ε v 0 strongly in L (0,T 0 ;H α ( for all α < s, where v 0 C 0 ([0,T 0 ];C 1 ( C 0,1 ([0,T 0 ];L 2 (.

17 ZERO-ELECTRON-MASS LIMIT IN THE HYDRODYNAMIC MODEL 17 It remains to show that v 0 is a solution of the incompressible Euler equations with amping. It hols for all χ C ([0,T 0 ], ψ C0 ( ; R such that ψ = 0, T0 (vt ε + v ε v ε + v ε ψχxt = 1 T0 (φ ε h ε ψχxt 0 T ε 2 0 = 1 T0 (φ ε h ε ψχxt = 0. ε 2 0 Letting ε 0 in the above equation gives (v 0 ψχ t + v 0 v 0 ψχ + v 0 ψχxt = 0. Q T0 By the efinition of weak erivatives, we conclue that v 0 t = P(v 0 v 0 + v 0, where P is the stanar projection on the set of ivergence-free vector fiels. Since we alreay have v 0 C 0 ([0,T 0 ];C 1 ( L (0,T 0 ;H s (, which implies that v 0 v 0 +v 0 C 0 ([0,T 0 ];C 0 ( L (0,T 0 ;H s 1 (, we infer v 0 t C 0 ( [0,T 0 ] L (0,T 0 ;H s 1 (. Thus, v 0 C 1 ( [0,T 0 ] is a classical solution to v 0 = 0, P(v 0 t + v 0 v 0 + v 0 = 0, v 0 (x, 0 = v 0 I(x, x, t > 0. The secon equation an the regularity of v 0 t +v 0 v 0 +v 0 show that there exists a function π L (0,T 0 ;H s ( such that v 0 = 0, v 0 t + v 0 v 0 + v 0 = π. Taking into account the equation satisfie by v ε an v ε t + v ε v ε + v ε v 0 t + v 0 v 0 + v 0 = π weakly* in L (0,T 0 ;L 2 (, we infer that 1 ε (φε h ε = v ε t + v ε v ε + v ε π weakly* in L (0,T 0 ;L 2 (. Finally, the uniqueness of smooth solutions to the incompressible Euler equations with amping implies the convergence of the whole sequences. This completes the proof. References [1] G. Alì, D. Bini an S. Rionero, Global existence an relaxation limit for smooth solutions to the Euler-Poisson moel for semiconuctors, SIAM J. Appl. Math. 32 (2000, [2] G. Alì an A. Jüngel, Global smooth solutions to the multi-imensional hyroynamic moel for two-carrier plasmas, J. Diff. Eqns. 190 (2003, [3] K. Asano, On the incompressible limit of the compressible Euler equations, Japan J. Appl. Math. 4 (1987, [4] S. Corier an E. Grenier, Quasineutral limit of an Euler-Poisson system arising from plasma physics, Comm. Part. Diff. Eqns. 25 (2000,

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19 ZERO-ELECTRON-MASS LIMIT IN THE HYDRODYNAMIC MODEL 19 [30] S. Wang, Z.P. Xin an P. Markowich, Quasineutral limit of the rift-iffusion moel for semiconuctors : the case of general sign-changing oping profile, SIAM J. Math. Anal. 37 (2006, Department of Mathematics, University of Calabria, an INFN-Gruppo c. Cosenza, Arcavacata i Rene (CS, Italy aress: Giuseppe.ali@unical.it Department of Mathematical Sciences, Tsinghua University, Beijing, , People s Republic of China aress: lchen@math.tsinghua.eu.cn Institute für Analysis un Scientific Computing, Technische Universität Wien, 1040 Wien, Austria aress: juengel@anum.tuwien.ac.at Laboratoire e Mathématiques, UMR 6620, CNRS-Université Blaise Pascal (Clermont- Ferran 2, Aubière, France aress: peng@math.univ-bpclermont.fr